# Chapter 7 Section 6

```Review Videos
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Graphing the x and y intercept
Graphing the x and y intercepts
Graphing a line in slope intercept form
Converting into slope intercept form
Chapter 7 Section 6
Families of Linear Graphs
What You’ll Learn
You’ll learn to explore the effects of changing
the slopes and y-intercepts of linear
functions.
Why It’s Important
Families of graphs can display different fees.
Families of linear graphs often fall into two
categories1. Those with the same slope
2. Those with the same y-intercept.
Family of Graphs
What do these lines
have in common?
y = &frac12;x + 3
y = &frac12;x - 1
Same Slope
Family of Graphs
What do these lines
have in common?
y = -x + 1
Same y-intercept
y = ⅓x + 1
Not a Family of Graphs
What do these lines
have in common?
y=x+2
y = ⅓x
Different
y-intercept and
slope
Example 1
Graph each pair of equations. Describe any similarities or
differences. Explain why they are a family of graphs.
y = 3x + 4
y = 3x – 2
The graphs have y-intercepts
y = 3x + 4
of 4 and -2, respectively.
They are a family of graphs
because the slope of each
line is 3.
y = 3x - 2
Example 2
Graph each pair of equations. Describe any similarities or
differences. Explain why they are a family of graphs.
y=x+3
y = -&frac12;x + 3
Each graph has a different slope. y = -&frac12;x + 3
Each graph has a y-intercept of 3.
Thus, they are a family of graphs.
y=x+3
Hint:
You can compare graphs of lines by looking at
their equations.
Example 3
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Matthew and Juan are starting their own pet care business. Juan wants to
charge \$5 an hour. Matthew thinks they should charge \$3 an hour. Suppose x
represents the number of hours. Then y = 5x and y = 3x represents how much
they would charge, respectively. Compare and contrast the graphs of the
equations.
6
The equations have the same yintercept, but the graph of y = 5x
is steeper. This is because its
slope, which represents \$5 per
hour, is greater that the slope of
the graph of y = 3x.
5
y = 5x
y = 3x
4
3
2
0
.5
1
1.5
2
Compare and contrast the graphs of the equations.
Verify by graphing the equation.
y = -3x + 4
y = -x + 4
y = -3x + 4
y = -x + 4
Same y-intercept
Different slope
y = -3x + 4
y = -x + 4
Try This One
Compare and contrast the graphs of the equations.
Verify by graphing the equation.
y = ⅔x + 3
y = ⅔x -1
y = ⅔x + 3
y = ⅔x -1
y = ⅔x + 3
Same slope
Different y-intercept
y = ⅔x - 1
Parent Graph
A parent graph is the simplest of the graphs in a family. Let’s
summarize how changing the m or b in y = mx + b affects the
graph of the equation.
Parent: y = x
As the value of m
Increases, the line
Gets steeper
y = 3x
y=x
y = &frac14;x
Parent: y = -x
y = -3x
y = -x
As the value of m
Decreases, the line
Gets steeper.
y = -&frac14;x
Parent: y = 2x
y = 2x
y = 2x + 3
y = 2x - 4
As the value of b
Increases, the graph shifts
Up on the y-axis.
As the value of b decreases,
the graph shifts down on the
y-axis
You can change a graph by changing the slope or y-intercept.
Example 4
Change y = -&frac12;x + 3 so that the graph of the new equation fits
each description.
y = -2x + 3
Same y-intercept, steeper negative slope
The y-intercept is 3, and the slope is -&frac12;. The
new equation will also have a y-intercept
of 3. In order for the slope to be steeper
and still be negative, its value must be less
than -&frac12;, such as -2. The new equation is
y = -2x + 3.
y = -&frac12;x + 3
Example 5
Change y = -&frac12;x + 3 so that the graph of the new equation fits
each description.
y = -&frac12;x + 7
Same slope, y-intercept is shifted up 4 units
The slope of the new equation will be -&frac12;.
Since the current y-intercept will be 3 + 4 or 7.
The new equation is y = -&frac12;x + 7.
Always check by graphing
y = -&frac12;x + 3
Change y = 2x + 1 so that the graph of the new
equation fits each description.
Same slope, shifted
down 1 unit.
y = 2x + 0
Simplified to y = 2x